APPROXIMATE ANALYTICAL SOLUTION OF NON-LINEAR BOUSSINESQ EQUATION FOR THE UNSTEADY GROUND WATER FLOW IN AN UNCONFINED AQUIFER BY HOMOTOPY PERTURBATION TRANSFORM METHOD
For one dimensional homogeneous, isotropic aquifer, without accretion the governing Boussinesq
equation under Dupuit assumptions is a nonlinear partial differential equation. In the present paper
approximate analytical solution of nonlinear Boussinesq equation is obtained using Homotopy
perturbation transform method(HPTM). The solution is compared with the exact solution. The
comparison shows that the HPTM is efficient, accurate and reliable. The analysis of two important aquifer
parameters namely viz. specific yield and hydraulic conductivity is studied to see the effects on the height
of water table. The results resemble well with the physical phenomena.
Numerical simulation on laminar free convection flow and heat transfer over a...eSAT Journals
Abstract
In the present numerical study, laminar free-convection flow and heat transfer over an isothermal vertical plate is presented. By
means of similarity transformation, the original nonlinear coupled partial differential equations of flow are transformed to a pair
of simultaneous nonlinear ordinary differential equations. Then, they are reduced to first order system. Finally, NewtonRaphson
method and adaptive Runge-Kutta method are used for their integration. The computer codes are developed for this numerical
analysis in Matlab environment. Velocity and temperature profiles for various Prandtl number are illustrated graphically. Flow
and heat transfer parameters are derived as functions of Prandtl number alone. The results of the present simulation are then
compared with experimental data published in literature and find a good agreement.
Keywords: Free Convection, Heat Transfer, Matlab, Numerical Simulation, Vertical Plate.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
Some new exact Solutions for the nonlinear schrödinger equationinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Existence of Solutions of Fractional Neutral Integrodifferential Equations wi...inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Numerical simulation on laminar free convection flow and heat transfer over a...eSAT Journals
Abstract
In the present numerical study, laminar free-convection flow and heat transfer over an isothermal vertical plate is presented. By
means of similarity transformation, the original nonlinear coupled partial differential equations of flow are transformed to a pair
of simultaneous nonlinear ordinary differential equations. Then, they are reduced to first order system. Finally, NewtonRaphson
method and adaptive Runge-Kutta method are used for their integration. The computer codes are developed for this numerical
analysis in Matlab environment. Velocity and temperature profiles for various Prandtl number are illustrated graphically. Flow
and heat transfer parameters are derived as functions of Prandtl number alone. The results of the present simulation are then
compared with experimental data published in literature and find a good agreement.
Keywords: Free Convection, Heat Transfer, Matlab, Numerical Simulation, Vertical Plate.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
Some new exact Solutions for the nonlinear schrödinger equationinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Existence of Solutions of Fractional Neutral Integrodifferential Equations wi...inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
A Tau Approach for Solving Fractional Diffusion Equations using Legendre-Cheb...iosrjce
In this paper, a modified numerical algorithm for solving the fractional diffusion equation is
proposed. Based on Tau idea where the shifted Legendre polynomials in time and the shifted Chebyshev
polynomials in space are utilized respectively.
The problem is reduced to the solution of a system of linear algebraic equations. From the computational point
of view, the solution obtained by this approach is tested and the efficiency of the proposed method is confirmed.
A high accuracy approximation for half - space problems with anisotropic scat...IOSR Journals
An approximate model, which is developed previously, is extended to solve the half – space problems
in the case of extremely anisotropic scattering kernels. The scattering kernel is assumed to be a combination of
isotropic plus a forward and backward leak. The transport equation is transformed into an equivalent fictitious
one involving only multiple isotropic scattering, therefore permitting the application of the previously developed
method for treating isotropic scattering. It has been shown that the method solves the albedo half – space
problem in a concise manner and leads to fast converging numerical results as shown in the Tables. For pure
scattering and weakly absorbing medium the computations can be performed by hand with a pocket calculator
Exact Solutions for MHD Flow of a Viscoelastic Fluid with the Fractional Bur...IJMER
This paper presents an analytical study for the magnetohydrodynamic (MHD) flow of a
generalized Burgers’ fluid in an annular pipe. Closed from solutions for velocity is obtained by using finite
Hankel transform and discrete Laplace transform of the sequential fractional derivatives. Finally, the
figures are plotted to show the effects of different parameters on the velocity profile.
A generalized bernoulli sub-ODE Method and Its applications for nonlinear evo...inventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
SCHRODINGER'S CAT PARADOX RESOLUTION USING GRW COLLAPSE MODELijrap
Possible solution of the Schrödinger's cat paradox is considered.We pointed out that: the collapsed
state of the cat always shows definite and predictable measurement outcomes even if Schrödinger's
cat consists of a superposition: cat=livecat+
deathcat
In this talk, we discuss some recent advances in probabilistic schemes for high-dimensional PIDEs. It is known that traditional PDE solvers, e.g., finite element, finite difference methods, do not scale well with the increase of dimension. The idea of probabilistic schemes is to link a wide class of nonlinear parabolic PIDEs to stochastic Levy processes based on nonlinear version of the Feynman-Kac theory. As such, the solution of the PIDE can be represented by a conditional expectation (i.e., a high-dimensional integral) with respect to a stochastic dynamical system driven by Levy processes. In other words, we can solve the PIDEs by performing high-dimensional numerical integration. A variety of quadrature methods could be applied, including MC, QMC, sparse grids, etc. The probabilistic schemes have been used in many application problems, e.g., particle transport in plasmas (e.g., Vlasov-Fokker-Planck equations), nonlinear filtering (e.g., Zakai equations), and option pricing, etc.
Navier stokes equation in coordinates binormal, tangent and normalCarlos López
The Navier-Stokes problem is a very important set of partial differential equations for analyzing fluids into the context
of the motion of fluid substances. There is no a general analytical solution related to complex fields of velocity vector
푢(푋, 푡)
, wherein the position vector is given by 푋 = (푥, 푦. 푧) and 푡 is the time variable, but there are some few solutions
associated to the simple velocity vector and the pressure 푃(푋, 푡) experienced by the fluid. However, these simple
models are not sufficient to predict the dynamic of Newtonian fluids in general. On this article is proposed an
interesting mathematical model to represent easily the equations of Navier Stokes in a TNB frame system which let
optimize the task of modeling complex equations from a Cartesian coordinate system and reducing them to a set of
equations less complex in a TNB frame whose perspective is going to be truly interesting from the physical problem.
Sinc collocation linked with finite differences for Korteweg-de Vries Fraction...IJECEIAES
A novel numerical method is proposed for Korteweg-de Vries Fractional Equation. The fractional derivatives are described based on the Caputo sense. We construct the solution using different approach, that is based on using collocation techniques. The method combining a finite difference approach in the time-fractional direction, and the Sinc-Collocation in the space direction, where the derivatives are replaced by the necessary matrices, and a system of algebraic equations is obtained to approximate solution of the problem. The numerical results are shown to demonstrate the efficiency of the newly proposed method. Easy and economical implementation is the strength of this method.
Bound- State Solution of Schrodinger Equation with Hulthen Plus Generalized E...ijrap
We apply an approximation to centrifugal term to find bound state solutions to Schrodinger equation with
Hulthen Plus generalized exponential Coulomb potential Using Nikiforov-Uvarov Method. Using this
method, we obtained the energy-eigen value and the total wave function. We implement C++ algorithm, to
obtained the numerical values of the energy for different quantum state starting from the first excited state
for different values of the screening parameter.
Approximate Analytical Solution of Non-Linear Boussinesq Equation for the Uns...mathsjournal
For one dimensional homogeneous, isotropic aquifer, without accretion the governing Boussinesq equation under Dupuit assumptions is a nonlinear partial differential equation. In the present paper approximate analytical solution of nonlinear Boussinesq equation is obtained using Homotopy perturbation transform method(HPTM). The solution is compared with the exact solution. The comparison shows that the HPTM is efficient, accurate and reliable. The analysis of two important aquifer
parameters namely viz. specific yield and hydraulic conductivity is studied to see the effects on the height of water table. The results resemble well with the physical phenomena.
Decay Property for Solutions to Plate Type Equations with Variable CoefficientsEditor IJCATR
In this paper we consider the initial value problem for a plate type equation with variable coefficients and memory in
1 n R n ), which is of regularity-loss property. By using spectrally resolution, we study the pointwise estimates in the spectral
space of the fundamental solution to the corresponding linear problem. Appealing to this pointwise estimates, we obtain the global
existence and the decay estimates of solutions to the semilinear problem by employing the fixed point theorem
Applications of Homotopy perturbation Method and Sumudu Transform for Solving...IJRES Journal
In this paper, we make use of the properties of the Sumudu transform to find the exact solution of
fractional initial-boundary value problem (FIBVP) and fractional KDV equation. The method, namely,
homotopy perturbation Sumudu transform method, is the combination of the Sumudu transform and the
HPM using He’s polynomials. This method is very powerful and professional techniques for solving different
kinds of linear and nonlinear fractional differential equations arising in different fields of science and
engineering. We also present two different examples to illustrate the preciseness and effectiveness of this
method.
SUCCESSIVE LINEARIZATION SOLUTION OF A BOUNDARY LAYER CONVECTIVE HEAT TRANSFE...ijcsa
The purpose of this paper is to discuss the flow of forced convection over a flat plate. The governing partial
differential equations are transformed into ordinary differential equations using suitable transformations.
The resulting equations were solved using a recent semi-numerical scheme known as the successive
linearization method (SLM). A comparison between the obtained results with homotopy perturbation method and numerical method (NM) has been included to test the accuracy and convergence of the method.
A solution of the Burger’s equation arising in the Longitudinal Dispersion Ph...IOSR Journals
The goal of the paper is to examine the concentration of the longitudinal dispersion phenomenon arising in fluid flow through porous media. The solution of the Burger’s equation for the dispersion problem is
presented by approach to Sumudu transformation. The solution is obtained by using suitable conditions and is
more simplified under the standard assumptions.
A Tau Approach for Solving Fractional Diffusion Equations using Legendre-Cheb...iosrjce
In this paper, a modified numerical algorithm for solving the fractional diffusion equation is
proposed. Based on Tau idea where the shifted Legendre polynomials in time and the shifted Chebyshev
polynomials in space are utilized respectively.
The problem is reduced to the solution of a system of linear algebraic equations. From the computational point
of view, the solution obtained by this approach is tested and the efficiency of the proposed method is confirmed.
A high accuracy approximation for half - space problems with anisotropic scat...IOSR Journals
An approximate model, which is developed previously, is extended to solve the half – space problems
in the case of extremely anisotropic scattering kernels. The scattering kernel is assumed to be a combination of
isotropic plus a forward and backward leak. The transport equation is transformed into an equivalent fictitious
one involving only multiple isotropic scattering, therefore permitting the application of the previously developed
method for treating isotropic scattering. It has been shown that the method solves the albedo half – space
problem in a concise manner and leads to fast converging numerical results as shown in the Tables. For pure
scattering and weakly absorbing medium the computations can be performed by hand with a pocket calculator
Exact Solutions for MHD Flow of a Viscoelastic Fluid with the Fractional Bur...IJMER
This paper presents an analytical study for the magnetohydrodynamic (MHD) flow of a
generalized Burgers’ fluid in an annular pipe. Closed from solutions for velocity is obtained by using finite
Hankel transform and discrete Laplace transform of the sequential fractional derivatives. Finally, the
figures are plotted to show the effects of different parameters on the velocity profile.
A generalized bernoulli sub-ODE Method and Its applications for nonlinear evo...inventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
SCHRODINGER'S CAT PARADOX RESOLUTION USING GRW COLLAPSE MODELijrap
Possible solution of the Schrödinger's cat paradox is considered.We pointed out that: the collapsed
state of the cat always shows definite and predictable measurement outcomes even if Schrödinger's
cat consists of a superposition: cat=livecat+
deathcat
In this talk, we discuss some recent advances in probabilistic schemes for high-dimensional PIDEs. It is known that traditional PDE solvers, e.g., finite element, finite difference methods, do not scale well with the increase of dimension. The idea of probabilistic schemes is to link a wide class of nonlinear parabolic PIDEs to stochastic Levy processes based on nonlinear version of the Feynman-Kac theory. As such, the solution of the PIDE can be represented by a conditional expectation (i.e., a high-dimensional integral) with respect to a stochastic dynamical system driven by Levy processes. In other words, we can solve the PIDEs by performing high-dimensional numerical integration. A variety of quadrature methods could be applied, including MC, QMC, sparse grids, etc. The probabilistic schemes have been used in many application problems, e.g., particle transport in plasmas (e.g., Vlasov-Fokker-Planck equations), nonlinear filtering (e.g., Zakai equations), and option pricing, etc.
Navier stokes equation in coordinates binormal, tangent and normalCarlos López
The Navier-Stokes problem is a very important set of partial differential equations for analyzing fluids into the context
of the motion of fluid substances. There is no a general analytical solution related to complex fields of velocity vector
푢(푋, 푡)
, wherein the position vector is given by 푋 = (푥, 푦. 푧) and 푡 is the time variable, but there are some few solutions
associated to the simple velocity vector and the pressure 푃(푋, 푡) experienced by the fluid. However, these simple
models are not sufficient to predict the dynamic of Newtonian fluids in general. On this article is proposed an
interesting mathematical model to represent easily the equations of Navier Stokes in a TNB frame system which let
optimize the task of modeling complex equations from a Cartesian coordinate system and reducing them to a set of
equations less complex in a TNB frame whose perspective is going to be truly interesting from the physical problem.
Sinc collocation linked with finite differences for Korteweg-de Vries Fraction...IJECEIAES
A novel numerical method is proposed for Korteweg-de Vries Fractional Equation. The fractional derivatives are described based on the Caputo sense. We construct the solution using different approach, that is based on using collocation techniques. The method combining a finite difference approach in the time-fractional direction, and the Sinc-Collocation in the space direction, where the derivatives are replaced by the necessary matrices, and a system of algebraic equations is obtained to approximate solution of the problem. The numerical results are shown to demonstrate the efficiency of the newly proposed method. Easy and economical implementation is the strength of this method.
Bound- State Solution of Schrodinger Equation with Hulthen Plus Generalized E...ijrap
We apply an approximation to centrifugal term to find bound state solutions to Schrodinger equation with
Hulthen Plus generalized exponential Coulomb potential Using Nikiforov-Uvarov Method. Using this
method, we obtained the energy-eigen value and the total wave function. We implement C++ algorithm, to
obtained the numerical values of the energy for different quantum state starting from the first excited state
for different values of the screening parameter.
A numerical analysis of three dimensional darcy model in an inclined rect
Similar to APPROXIMATE ANALYTICAL SOLUTION OF NON-LINEAR BOUSSINESQ EQUATION FOR THE UNSTEADY GROUND WATER FLOW IN AN UNCONFINED AQUIFER BY HOMOTOPY PERTURBATION TRANSFORM METHOD
Approximate Analytical Solution of Non-Linear Boussinesq Equation for the Uns...mathsjournal
For one dimensional homogeneous, isotropic aquifer, without accretion the governing Boussinesq equation under Dupuit assumptions is a nonlinear partial differential equation. In the present paper approximate analytical solution of nonlinear Boussinesq equation is obtained using Homotopy perturbation transform method(HPTM). The solution is compared with the exact solution. The comparison shows that the HPTM is efficient, accurate and reliable. The analysis of two important aquifer
parameters namely viz. specific yield and hydraulic conductivity is studied to see the effects on the height of water table. The results resemble well with the physical phenomena.
Decay Property for Solutions to Plate Type Equations with Variable CoefficientsEditor IJCATR
In this paper we consider the initial value problem for a plate type equation with variable coefficients and memory in
1 n R n ), which is of regularity-loss property. By using spectrally resolution, we study the pointwise estimates in the spectral
space of the fundamental solution to the corresponding linear problem. Appealing to this pointwise estimates, we obtain the global
existence and the decay estimates of solutions to the semilinear problem by employing the fixed point theorem
Applications of Homotopy perturbation Method and Sumudu Transform for Solving...IJRES Journal
In this paper, we make use of the properties of the Sumudu transform to find the exact solution of
fractional initial-boundary value problem (FIBVP) and fractional KDV equation. The method, namely,
homotopy perturbation Sumudu transform method, is the combination of the Sumudu transform and the
HPM using He’s polynomials. This method is very powerful and professional techniques for solving different
kinds of linear and nonlinear fractional differential equations arising in different fields of science and
engineering. We also present two different examples to illustrate the preciseness and effectiveness of this
method.
SUCCESSIVE LINEARIZATION SOLUTION OF A BOUNDARY LAYER CONVECTIVE HEAT TRANSFE...ijcsa
The purpose of this paper is to discuss the flow of forced convection over a flat plate. The governing partial
differential equations are transformed into ordinary differential equations using suitable transformations.
The resulting equations were solved using a recent semi-numerical scheme known as the successive
linearization method (SLM). A comparison between the obtained results with homotopy perturbation method and numerical method (NM) has been included to test the accuracy and convergence of the method.
A solution of the Burger’s equation arising in the Longitudinal Dispersion Ph...IOSR Journals
The goal of the paper is to examine the concentration of the longitudinal dispersion phenomenon arising in fluid flow through porous media. The solution of the Burger’s equation for the dispersion problem is
presented by approach to Sumudu transformation. The solution is obtained by using suitable conditions and is
more simplified under the standard assumptions.
Numerical Solution of Diffusion Equation by Finite Difference Methodiosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Elzaki transform homotopy perturbation method for solving porous medium equat...eSAT Journals
Abstract In this paper, the ELzaki transform homotopy perturbation method (ETHPM) has been successfully applied to obtain the approximate analytical solution of the nonlinear homogeneous and non-homogeneous gas dynamics equations. The proposed method is an elegant combination of the new integral transform “ELzaki Transform” and the homotopy perturbation method. The method is really capable of reducing the size of the computational work besides being effective and convenient for solving nonlinear equations. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. A clear advantage of this technique over the decomposition method is that no calculation of Adomian’s polynomials is needed. Keywords: ELzaki transform, Homotopy perturbation method, non linear partial differential equation, and nonlinear gas dynamics equation
Elzaki transform homotopy perturbation method for solving porous medium equat...eSAT Journals
Abstract In this paper, we apply a new method called ELzaki transform homotopy perturbation method (ETHPM) to solve porous medium equation. This method is a combination of the new integral transform “ELzaki transform” and the homotopy perturbation method. The nonlinear term can be easily handled by homotopy perturbation method. The porous medium equations have importance in engineering and sciences and constitute a good model for many systems in various fields. Some cases of the porous medium equation are solved as examples to illustrate ability and reliability of mixture of ELzaki transform and homotopy perturbation method. The results reveal that the combination of ELzaki transform and homotopy perturbation method is quite capable, practically well appropriate for use in such problems and can be applied to other nonlinear problems. This method is seen as a better alternative method to some existing techniques for such realistic problems. Key words: ELzaki transform, Homotopy perturbation method, non linear partial differential equation, and porous medium equation
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
The numerical solution of helmholtz equation via multivariate padé approximationeSAT Journals
Abstract
In this study, we consider the numerical solution of the Helmholtz equation, arising from numerous physical phenomena and
engineering applications including energy systems. We make use of a Padé Approximation method for the solution of the
Helmholtz equation. Firstly, Helmholtz partial differential equation had been converted to power series by two-dimensional
differential transformation, then the numerical solution of the equation was put into Padé series form for accelerating
convergence and decreasing computational time. Hereby, we obtained numerical solution of Helmholtz type partial differential
equation.
Key Words: Helmholtz equation, Two-dimensional differential transformation, Multivariate Padé approximation,
power series
Pressure Gradient Influence on MHD Flow for Generalized Burgers’ Fluid with S...IJERA Editor
This paper presents a research for magnetohydrodynamic (MHD) flow of an incompressible generalized
Burgers’ fluid including by an accelerating plate and flowing under the action of pressure gradient. Where the
no – slip assumption between the wall and the fluid is no longer valid. The fractional calculus approach is
introduced to establish the constitutive relationship of the generalized Burgers’ fluid. By using the discrete
Laplace transform of the sequential fractional derivatives, a closed form solutions for the velocity and shear
stress are obtained in terms of Fox H- function for the following two problems: (i) flow due to a constant
pressure gradient, and (ii) flow due to due to a sinusoidal pressure gradient. The solutions for no – slip condition
and no magnetic field, can be derived as special cases of our solutions. Furthermore, the effects of various
parameters on the velocity distribution characteristics are analyzed and discussed in detail. Comparison between
the two cases is also made.
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
FITTED OPERATOR FINITE DIFFERENCE METHOD FOR SINGULARLY PERTURBED PARABOLIC C...ieijjournal
In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type
with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson
extrapolation method is applied on the time direction and the fitted operator finite difference method on the
spatial direction is used, on the uniform grids. The stability and consistency of the method were established
very well to guarantee the convergence of the method. Numerical experimentation is carried out on model
examples, and the results are presented both in tables and graphs. Further, the present method gives a more
accurate solution than some existing methods reported in the literature.
Similar to APPROXIMATE ANALYTICAL SOLUTION OF NON-LINEAR BOUSSINESQ EQUATION FOR THE UNSTEADY GROUND WATER FLOW IN AN UNCONFINED AQUIFER BY HOMOTOPY PERTURBATION TRANSFORM METHOD (20)
OPTIMIZING SIMILARITY THRESHOLD FOR ABSTRACT SIMILARITY METRIC IN SPEECH DIAR...mathsjournal
Speaker diarization is a critical task in speech processing that aims to identify "who spoke when?" in an
audio or video recording that contains unknown amounts of speech from unknown speakers and unknown
number of speakers. Diarization has numerous applications in speech recognition, speaker identification,
and automatic captioning. Supervised and unsupervised algorithms are used to address speaker diarization
problems, but providing exhaustive labeling for the training dataset can become costly in supervised
learning, while accuracy can be compromised when using unsupervised approaches. This paper presents a
novel approach to speaker diarization, which defines loosely labeled data and employs x-vector embedding
and a formalized approach for threshold searching with a given abstract similarity metric to cluster
temporal segments into unique user segments. The proposed algorithm uses concepts of graph theory,
matrix algebra, and genetic algorithm to formulate and solve the optimization problem. Additionally, the
algorithm is applied to English, Spanish, and Chinese audios, and the performance is evaluated using wellknown similarity metrics. The results demonstrate that the robustness of the proposed approach. The
findings of this research have significant implications for speech processing, speaker identification
including those with tonal differences. The proposed method offers a practical and efficient solution for
speaker diarization in real-world scenarios where there are labeling time and cost constraints.
A POSSIBLE RESOLUTION TO HILBERT’S FIRST PROBLEM BY APPLYING CANTOR’S DIAGONA...mathsjournal
We present herein a new approach to the Continuum hypothesis CH. We will employ a string conditioning,
a technique that limits the range of a string over some of its sub-domains for forming subsets K of R. We
will prove that these are well defined and in fact proper subsets of R by making use of Cantor’s Diagonal
argument in its original form to establish the cardinality of K between that of (N,R) respectively
A Positive Integer 𝑵 Such That 𝒑𝒏 + 𝒑𝒏+𝟑 ~ 𝒑𝒏+𝟏 + 𝒑𝒏+𝟐 For All 𝒏 ≥ 𝑵mathsjournal
According to Bertrand's postulate, we have 𝑝𝑛 + 𝑝𝑛 ≥ 𝑝𝑛+1. Is it true that for all 𝑛 > 1 then 𝑝𝑛−1 + 𝑝𝑛 ≥𝑝𝑛+1? Then 𝑝𝑛 + 𝑝𝑛+3 > 𝑝𝑛+1 + 𝑝𝑛+2where 𝑛 ≥ 𝑁, 𝑁 is a large enough value?
A POSSIBLE RESOLUTION TO HILBERT’S FIRST PROBLEM BY APPLYING CANTOR’S DIAGONA...mathsjournal
We present herein a new approach to the Continuum hypothesis CH. We will employ a string conditioning,
a technique that limits the range of a string over some of its sub-domains for forming subsets K of R. We
will prove that these are well defined and in fact proper subsets of R by making use of Cantor’s Diagonal
argument in its original form to establish the cardinality of K between that of (N,R) respectively.
Moving Target Detection Using CA, SO and GO-CFAR detectors in Nonhomogeneous ...mathsjournal
systems in complex situations. A fundamental problem in radar systems is to automatically detect targets while maintaining a
desired constant false alarm probability. This work studies two detection approaches, the first with a fixed threshold and the
other with an adaptive one. In the latter, we have learned the three types of detectors CA, SO, and GO-CFAR. This research
aims to apply intelligent techniques to improve detection performance in a nonhomogeneous environment using standard
CFAR detectors. The objective is to maintain the false alarm probability and enhance target detection by combining
intelligent techniques. With these objectives in mind, implementing standard CFAR detectors is applied to nonhomogeneous
environment data. The primary focus is understanding the reason for the false detection when applying standard CFAR
detectors in a nonhomogeneous environment and how to avoid it using intelligent approaches.
OPTIMIZING SIMILARITY THRESHOLD FOR ABSTRACT SIMILARITY METRIC IN SPEECH DIAR...mathsjournal
Speaker diarization is a critical task in speech processing that aims to identify "who spoke when?" in an
audio or video recording that contains unknown amounts of speech from unknown speakers and unknown
number of speakers. Diarization has numerous applications in speech recognition, speaker identification,
and automatic captioning. Supervised and unsupervised algorithms are used to address speaker diarization
problems, but providing exhaustive labeling for the training dataset can become costly in supervised
learning, while accuracy can be compromised when using unsupervised approaches. This paper presents a
novel approach to speaker diarization, which defines loosely labeled data and employs x-vector embedding
and a formalized approach for threshold searching with a given abstract similarity metric to cluster
temporal segments into unique user segments. The proposed algorithm uses concepts of graph theory,
matrix algebra, and genetic algorithm to formulate and solve the optimization problem. Additionally, the
algorithm is applied to English, Spanish, and Chinese audios, and the performance is evaluated using wellknown similarity metrics. The results demonstrate that the robustness of the proposed approach. The
findings of this research have significant implications for speech processing, speaker identification
including those with tonal differences. The proposed method offers a practical and efficient solution for
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• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
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APPROXIMATE ANALYTICAL SOLUTION OF NON-LINEAR BOUSSINESQ EQUATION FOR THE UNSTEADY GROUND WATER FLOW IN AN UNCONFINED AQUIFER BY HOMOTOPY PERTURBATION TRANSFORM METHOD
1. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 2, June 2015
1
APPROXIMATE ANALYTICAL SOLUTION OF
NON-LINEAR BOUSSINESQ EQUATION
FOR THE UNSTEADY GROUND WATER
FLOW IN AN UNCONFINED AQUIFER BY
HOMOTOPY PERTURBATION TRANSFORM
METHOD
Deepti Mishraa
, Vikas Pradhanb
and Manoj Mehtab
a
LDRP Institute of Technology & Research , Ganddhinagar, India
b
Department of Applied Mathematics & Humanities, S.V.N.I.T, Surat 395007, India
Abstract
For one dimensional homogeneous, isotropic aquifer, without accretion the governing Boussinesq
equation under Dupuit assumptions is a nonlinear partial differential equation. In the present paper
approximate analytical solution of nonlinear Boussinesq equation is obtained using Homotopy
perturbation transform method(HPTM). The solution is compared with the exact solution. The
comparison shows that the HPTM is efficient, accurate and reliable. The analysis of two important aquifer
parameters namely viz. specific yield and hydraulic conductivity is studied to see the effects on the height
of water table. The results resemble well with the physical phenomena.
Keywords
Aquifers, stream-aquifer interaction, Dupuit assumptions, Boussinesq equation, Homotopy perturbation
transform method.
1.Introduction
Studies[9]have shown that the total amount of surface and ground water is not enough for all
the demand of cities and agricultural areas especially those arid or semi-arid areas, due to the
development of industry, agriculture and the increase of the population. The conjunctive
management of surface and ground water is a subject of great practical, economic and political
importance in the field of water resources. Numerous developments associated with this subject
have been obtained during last two decades. An alternative management strategy is to use
aquifers, the natural underground reservoirs which contain ten or hundred times more water than
is held in storage in a river or in surface reservoirs. These underground reservoirs are naturally to
filtering water and regulating water to some degree. Large amounts of water from precipitation
or irrigation percolate down into water table as an input to aquifer. The earliest study on
the interaction of river and aquifer was developed by [11]. He derived an analytical solution
for estimation of the flow from a stream to an aquifer caused by pumping near the stream.
Serrano (1998) presented a numerical model for transient stream/aquifer interactions in an
alluvial valley aquifer [8]. The model is based on the one-dimensional Boussinesq equation for
horizontal unconfined aquifer which was solved using a decomposition method. Parlange et
2. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 2, June 2015
2
al. (2001) developed a numerical model based on the one-dimensional Boussinesq equation for
horizontal unconfined aquifer and obtain its solution by solving using finite element program.
Verhoest et al. (2002) presented a numerical model and compared the results against a transient
analytical solution. Recently in [5] Hernandez and Uddameri have developed a semi-analytical
solution for stream-aquifer interactions under triangular stream-stage variations. In the present
paper it is assumed that the aquifer is in contact with a drainage canal at one end of the
horizontal aquifer and it is bounded by a zero flux at the impervious surface at the other end of
aquifer. For one dimensional homogeneous, isotropic aquifer, without accretion the governing
Boussinesq equation under Dupuit assumptions is a nonlinear partial differential equation. For
specific initial and boundary conditions the nonlinear Boussinesq equation is solved using
Homotopy perturbation transform method. The analysis of various parameters is studied to
observe the corresponding effect on the height of water table.
2. Analysis of Homotopy perturbation transform method
We present a Homotopy perturbation transform algorithm [1,6,10] for solving partial differential
equation written in an operator form
( , ) ( , ) ( , ) ( , )Du x t Ru x t Nu x t g x t (1)
with the initial conditions
( ,0) ( ), ( ,0) ( )tu x h x u x f x
where D is the second order linear differential operator
2
2
D
t
, R is the linear differential
operator of less order than D , N represents the general non-linear differential operator
and ( , )g x t is the source term. The method consist of first applying the Laplace transform to
equation (1) and then by using initial conditions, we have
[ ( , )] [ ( , )] [ ( , )] [ ( , )]L Du x t L Ru x t L Nu x t L g x t (2)
Using Laplace transform of derivatives and applying initial conditions, we have
2
[ ( , ) ,0 ,0 ] [ ( , )] [ ( , )] [ ( , )]ts u x s su x u x L Ru x t L Nu x t L g x t (3)
On simplifying
2 2 2 2
( ) ( ) 1 1 1
( , ) [ ( , )] [ ( , )] [ ( , )]
h x f x
u x s L Ru x t L g x t L Nu x t
s s s s s
(4)
Applying inverse Laplace transform in equation (4), we get
1
2
1
( , ) ( , ) ( , ) ( , )u x t G x t L L Ru x t Nu x t
s
(5)
3. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 2, June 2015
3
where ,G x t represents the term arising from source term and prescribed initial conditions.
Now, we apply the homotopy perturbation method according to which u can be expanded into
infinite series as
0
( , ) ( , )n
n
n
u x t p u x t
, (6)
where 0,1p is an embedding parameter. And the nonlinear term can be decomposed as
0
( , ) ( )n
n
n
Nu x t p H u
, (7)
where ( )nH u is the He’s polynomials can be generated by several means. Here we used the
following recursive formulation:
0
0 0
1
( ...... ) , 0,1,2,3,.......
!
n
i
n n in
i p
H u u N p u n
n p
(8)
By substituting equation (6) and (7) in equation (5) the solution can be written as
1
2
0 0 0
1
( , ) ( , ) ( , ) ( )n n n
n n n
n n n
p u x t G x t p L L R p u x t p H u
s
(9)
Equating the terms with identical power of p in equation (9), we obtained the following
approximations.
0
0: ( , ) ,p u x t G x t ,
1 1
1 0 02
1
: ( , ) ( , ) ( )p u x t L L Ru x t H u
s
,
2 1
2 1 12
1
: ( , ) ( , ) ( )p u x t L L Ru x t H u
s
, (10)
3 1
3 2 22
1
: ( , ) ( , ) ( )p u x t L L Ru x t H u
s
.
The best approximations for the solutions are
0 1 2
1
lim ...n
p
u u u u u
(11)
The present method finds the solution without any discretization or restrictive assumptions and
therefore reduces the numerical computation to a great extent.
4. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 2, June 2015
4
3.The nonlinear Boussinesq equation
The idealized cross section of the model under consideration is depicted in figure1 below.
Figure1. Idealized cross section of the model of transient stream-aquifer interaction.
The governing equation for one-dimensional, lateral, unconfined groundwater flow with Dupuit
assumptions is the Boussinesq equation (Bear, 1972) [2]:
y
h K h
h
t S x x
, (12)
where ,h x t is the water table elevation at a distance x from the origin and time t , K , yS are
the saturated hydraulic conductivity and specific yield respectively which are considered to be
constant.
In order to solve Boussinesq equation (12) completely, the specific initial condition and boundary
conditions are as considered in [8] are given by:
0, ( )h t H t , 0t , (13)
,
0xh l t
x
, 0t , (14)
The exact solution of equation(12) – (14) is as given in [7]
2
2/33
( , ) ( 1) 1
1 6( 1) 2( 1)
x x
h x t t
t t t
(15)
we have considered the case where ( )H t increases from zero to a maximum and then decreases
back to zero, thus providing a realistic behaviour for all times.
We take
2 33
( ) 1 1
2 1
H t t
t
shown in figure 2 below.
5. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 2, June 2015
5
0 5 10 15 20 25
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
H(t)
Time t
Figure 2.Schematic representation of H t .
H t reaches a maximum equal to
1
3
at
3
1
23t
. The interest of (13) is that the exact
solution given by (15) is available for comparison. In the boundary condition(14) zero flux is
considered at the impervious surface x = lx .
The initial condition is expressed by
0,0h x H x (16)
We assume initial water table in the aquifer as a quadratic approximation and choose
2
0H x a x bx c .Under the above initial and boundary conditions, the solution of
Boussinesq equation (12) is obtained by using Homotopy perturbation transform method.
4.Approximate analytical solution of Boussinesq equation for horizontal
aquifer
To solve equation (3.28) by applying Homotopy perturbation transform method the first step is to
apply Laplace transform on equation (12).
22
2
y
h K h h
L L h
t S xx
(17)
This can be written as
22
2
[ , ,0 ]
y
K h h
sh x s h x L h
S xx
(18)
On applying the initial condition (16), we get
6. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 2, June 2015
6
22
2
2
[ , ]
y
K h h
sh x s a x b x c L h
S xx
(19)
Taking Inverse Laplace transform, we get,
22
1 1 2 1
2
1 1
( , )
y
K h h
L h x s L a x b x c L L h
s S s xx
(20)
22
2 1
2
1
( , )
y
K h h
h x t a x b x c L L h
S s xx
(21)
Now we apply the Homotopy perturbation method to handle the non-linearity on the right hand
side of above equation, in the form
0
( , ) ,n
n
n
h x t p h x t
(22)
Using Binomial expansion and He’s Approximation, equation (3.37) reduces to
0 0
2 1
2
0
0
, ,
1
,
,
n n
n n
n n xxn
n
yn
n
n
n x
p h x t p h x t
K
p h x t a x bx c p L L
S s
p h x t
(23)
This can be written in expanded form as
2 2 2
2 20 1 2
0 1 2 2 2 2
2 2 1
0 1 2 22 2 2
20 1 2
2 2 2
( ..) ...
1
....
...
y
h h h
h ph p h p p
x x xK
h ph p h ax bx c p L L
S s h h h
p p
x x x
(24)
On comparing the coefficient of various power of p, we get
0 2
0: ( , )p h x t a x bx c
22
1 1 0 0
1 0 2
1
: ( , )
y
h hK
p h x t L L h
S s xx
22
2 1 0 01 1
2 0 12 2
1
: ( , ) 2
y
h hK h h
p h x t L L h h
S s x xx x
(25)
7. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 2, June 2015
7
222 2
3 1 0 02 1 2 1
3 0 1 22 2 2
1
: ( , ) 2
y
h hK h h h h
p h x t L L h h h
S s x x xx x x
2 22 2
4 1 3 0 0 32 1 1 2
4 0 1 2 32 2 2 2
1
: ( , ) 2 2
y
h h h hK h h h h
p h x t L L h h h h
S s x x x xx x x x
Proceeding in similar manner we can obtain further approximations. On solving above equations
and substituting in equation (3.38) we get HPTM solution of equation (3.28) in the form of a
series.
2
2 2
2
2 2 2 2
2
3
2 2 2 2 3
3
4
2 2 2 3 4
4
,
2 2
7 8 36 36
67 56 324 324 2
3
52 35 243 243 16
3
y
y
y
y
h x t ax bx c
K
b ax a ax bx c t
S
K
b ac abx a x at
S
K
b ac abx a x a t
S
K
b ac abx a x a t
S
(26)
Considering the initial depth at x = 0 to be zero, we choose c = 0. Applying boundary conditions
(3.29) and (3.30), we get 1 6a and 1b . On substituting the value of a and b and
assuming ratio 1yK S in equation (26) we get
22 2 2 3
2 2
4
2
1
, 1 7 6 67 54 9
6 3 3 6 6 54
2 81 27
52
81 2 4
x x x t t
h x t x t x x x x x
t
x x
(27)
On expansion it can be written as
2 3 4 2 2 2 2 2 3 2 4
2 3 47 67 104
,
6 54 81 6 6 6 6 6
t t t x x t x t x t x t
h x t t x xt xt xt xt (28)
On adding and subtracting a term
2 3 4
9 9 9 9 9
6 6 6 6 6
t t t t
in equation (28), we get
2 2 2 2 2 3 2 4 2 3 4
2 3 4
2 3 4 2 3 4
9 9 9 9 9
,
6 6 6 6 6 6 6 6 6 6
7 67 104 9 9 9 9 9
6 54 81 6 6 6 6 6
x x t x t x t x t t t t t
h x t x xt xt xt xt
t t t t t t t
t
(29)
8. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 2, June 2015
8
which can be written after rearranging the terms as,
2
1
3
3 3
,
6 1 2 1
x
h x t
t t
(30)
Equation (30) is the required solution of (12) obtained by using Homotopy perturbation transform
method which satisfies boundary conditions (13) and (14).
5.Results and Discussion
The numerical values obtained by HPTM for specific values of time and various space
coordinates are compared with the exact solution for lx =3(m). From the table 1, it can be seen
that the solution obtained by HPTM is very close to the exact solution for t = 0.25day. Its
graphical representation of comparison with exact solution is given in figure 3. The numerical
values at various values of time(days) and space(m) are shown in table 2. It has been observed
that the numerical values obtained by HPTM converge with the exact solution. From the table 2,
it is observed that the height of the water table at x = 0 increases with the time to its maximum
value
1
3
upto t = 4.2(days) and then decreases to zero after a very long time (approximately
4000days ).Also, at x = 3(m), the height of water table decreases from its initial value 1.5(m) to
0.8658(m) at t = 4.2(days) and becomes zero after a very long time (approximately 4000days). Its
graphical representation is shown in figure 4 and figure 5. The decrease of the height of the water
table at x = 0 is not shown in figure 4. Thus, the result satisfies the boundary conditions and
behaves well with the physical phenomena for various values of time.
Table 1.Numerical values of height of water table in an unconfined horizontal aquifer obtained by Exact
solution [4] and Homotopy perturbation transform method.
Distance (x) (m) ,h x t at (t =0.25days)
___________________________________________________________________
EXACT HPTM
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
0.19247665008383383
0.34714331675050053
0.4911433167505006
0.6244766500838338
0.7471433167505006
0.8591433167505006
0.9604766500838338
1.0511433167505002
1.1311433167505007
1.200476650083834
1.2591433167505004
1.3071433167505004
1.3444766500838339
1.3711433167505005
1.3871433167505005
1.3924766500838344
0.19247665008383374
0.3471433167505005
0.49114331675050027
0.6244766500838337
0.7471433167505003
0.8591433167505004
0.9604766500838338
1.0511433167505002
1.1311433167505003
1.2004766500838338
1.2591433167505004
1.3071433167505004
1.3444766500838337
1.3711433167505003
1.3871433167505003
1.3924766500838337
Table 2.Numerical values of water head in an unconfined horizontal aquifer for different distance x at
time t = 0,…,4.3 days.
10. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 2, June 2015
10
0 0.5 1 1.5 2 2.5 3
0
0.5
1
1.5
Distance x
heighth(x,t)
t = 0 days
t = 0.6 days
t = 1.2 days
t = 4.2 days
Figure 4.Height of water table in an unconfined horizontal aquifer for different distance x at time
t = 0,…,4.3 days.
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.5
1
1.5
Time t
heighth(x,t)
x = 0
x = 0.5
x = 1
x = 1.5
x = 2
x = 2.5
x = 3
Figure 5.Height of water table in an unconfined horizontal aquifer for different time t at distance x = 0 , 0.5
, 1,…, 3.
6.Sensitivity analysis
The effect of various parameters on the height of water table in an unconfined horizontal aquifer
has been observed and their numerical values and graphical representation are presented below.
The numerical values of ,h x t at fixed time t = 0.4(day) are observed for different distance x by
increasing the value of specific yield(Sy) keeping hydraulic conductivity(K) same. The numerical
values are shown in table 4. It can be seen that there is an effect on height of water with the
increase in Sy. From table 4 it is observed that with the least value of Sy(0.25) the height of the
water at x =3(m) is decreased from its initial value 1.5m to 1.025 where as with the highest value
of Sy(0.95), the height of the water table at x=3(m) is decreased from its initial value to
1.3402(m). A similar observation is made by keeping Sy fixed and increasing the various values
of K. The numerical values under this effect are shown in table 5. It is observed that with the least
value of K(1darcy) the height of the water table at x=3(m) is decreased from its initial value
1.5(m) to 1.0625(m)whereas with the highest value of K(1000darcy ) the height of the water table
at x=3(m) is decreased from its initial value to 0.3002(m). Thus, both the parameters are sensitive
with a small change in the values of Sy and higher change in K. The graphical representation of
11. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 2, June 2015
11
the effect of Sy and K are shown in figure 6 and figure 7 respectively. It is seen that with the
increasing value of Sy the height of the water is increases and with K the height of the water table
decreases which is consistent with respect to the properties of the aquifer parameters. The effect
of the ratio K/Sy is also observed and the numerical values of the height of water table in five
different materials are calculated at specific time(t=0.4day). The numerical values at time
t=0.4(day) are shown in table 6 and its graphical representation is shown is figure 8. From the
graphical representation it is seen that the height of the water table is maximum between x=0 to
x=3(m) in sandstone in comparison to other materials. From all the numerical values shown in
table 4 to table 6 it can be seen that the changes in the height of the water table with the space
variable is small. This is due to the fact that the groundwater flow is slow and aquifer does
behave as a reservoir to hold the water a relatively long time.[9]
Table 4.Numerical values of water head in an unconfined horizontal aquifer for different value of
yS Keeping 1 /K m day fixed at time t=0.4day.
Table 5.Numerical values of water head in an unconfined horizontal aquifer for different value
of K Keeping 0.21yS fixed at time t=0.4 day.
height ,h x t at 1 /K m day (t=0.4 day)
_______________________________________________________________________________________________________
x 0.25yS 0.5yS 0.75yS 0.95yS
0
0.5
1
1.5
2
2.5
3
0.2694
0.4405
0.5968
0.7382
0.8647
0.9763
1.073
0.2694
0.5071
0.7150
0.8931
1.0415
1.1601
1.2489
0.2694
0.5567
0.7993
0.9971
1.1501
1.2584
1.3219
0.2694
0.5893
0.8527
1.0594
1.2096
1.3032
1.3402
height ,h x t 0.21yS (t =0.4 day)
____________________________________________________________________________________________________
x 1K m/day 50K m/day 100K m/day 1000K m/day
0
0.5
1
1.5
2
2.5
3
0.2694
0.4269
0.5718
0.7042
0.8242
0.9316
1.0265
0.2694
0.2924
0.3152
0.3377
0.3600
0.3820
0.4038
0.2694
0.2857
0.3019
0.3180
0.3339
0.3497
0.3654
0.2694
0.2746
0.2797
0.2849
0.2900
0.2951
0.3002
12. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 2, June 2015
12
Table 6.Numerical values of water head in an unconfined horizontal aquifer for different value
of ratio yK S [3] at time t=0.4 day.
,h x t (t=0.4 day)
___________________________________________________________________________________________________
x 1K 1K 100K 100K 1000K
0.02yS 0.01yS 0.44yS 0.16yS 0.13yS
Sandstone Fine Sand Sand & gravel Coarse Sand Gravel
0
0.5
1
1.5
2
2.5
3
0.2694
0.3193
0.3681
0.4156
0.4619
0.5071
0.5510
0.2694
0.3048
0.3397
0.3739
0.4075
0.4405
0.4730
0.2694
0.2930
0.3163
0.3393
0.3621
0.3846
0.4068
0.2694
0.2837
0.2978
0.3118
0.3258
0.3397
0.3534
0.2694
0.2735
0.2775
0.2816
0.2856
0.2897
0.2937
0 0.5 1 1.5 2 2.5 3
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Distance x
heighth(x,t)
Sy
= 0.01
Sy
= 0.02
Sy
= 0.75
Sy
= 0.95
Figure 6.Height of water table in an unconfined horizontal aquifer for different value of
yS Keeping 1 /K m day fixed at time t=0.4 day.
0 0.5 1 1.5 2 2.5 3
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
Distance x
heighth(x,t)
K = 1 m/day
K = 50 m/day
K = 100 m/day
K = 1000 m/day
Figure 7.Height of water table in an unconfined horizontal aquifer for different value of K
Keeping 0.21yS fixed at time t=0.4 day.
13. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 2, June 2015
13
0 0.5 1 1.5 2 2.5 3
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
Distance x
heighth(x,t)
Sandstone
Fine sand
Sand and gravel
Coarse sand
Gravel
Figure 8.Height of water table in an unconfined horizontal aquifer for different values of value
of ratio yK S at time t=0.4 day.
7.Conclusion
The approximate analytical solution of Boussinesq equation for horizontal aquifer is obtained by
applying HPTM and compared with the available exact solution for the horizontal aquifer. We
see that HPTM is easy, accurate and convenient. The combination of Homotopy perturbation
method and Laplace transform overcomes the restriction of Laplace transform method to solve
non-linear partial differential equation. The two important parameters viz. Hydraulic conductivity
and Specific yield( yS ) are considered in the present groundwater flow problem. The approximate
analytical solution is obtained by considering the ratio 1
y
K
S
. However, the sensitivity of these
parameters is studied for five different ratios of five different samples. It is concluded that among
the five samples considered the height of the water is maximum in sandstone. Various authors
have obtained the solution of Boussinesq with different boundary conditions. We have shown that
HPTM can be applied equally well with the suitable choice of initial condition.
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[8] Serrano S. E and Workman S.R 1998 Modelling transient stream/aquifer interaction with the non
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Authors
Deepti Mishra is presently working as an Assistant Professor at LDRP Institute of
Technology and Research, Gandhinagar, Gujarat in India. She obtained M.Sc. in applied
Mathematics in 2008 from M.S University, Vadodara in India. She awarded Ph.D in
Mathematics in June 2014 from Department of Applied Mathematics & Humanities, Sardar
Vallabhbhai National Institute of Technology, Surat in INDIA. Her current research
interest is fluid flow through porous media, mathematical modelling and approximate analytical techniques.
She has 6 years of teaching experience at undergraduate and postgraduate level. She has attended around 12
conferences at national and international level. She has published many research papers in various
international journals and presented around 5 papers at international conferences. She is member of reputed
societies like Indian society of theoretical and applied mechanics (ISTAM), Indian Mathematical Society
(IMS).
Vikas Pradhan is Associate Professor and Former Head of Department of Applied
Mathematics & Humanities, Sardar Vallabhbhai National Institute of Technology, Surat
in INDIA. He obtained Ph.D in fluid flow through porous media from South Gujarat
University. His areas of specialization are Fluid flow through porous media, Advanced
Analytical & Numerical Techniques. He has 24 years of teaching experience at
undergraduate & post Graduate level. He supervised three Ph.D, one M. Phil student and six M.Sc.
dissertations. At present 10 student are pursuing their research work under him. He is member of many
reputed societies like Indian society of theoretical and applied mechanics (ISTAM), Indian Mathematical
Society (IMS), Indian Society for Technical Education (ISTE), Gujarat Ganit Mandal (GGM). He was
former vice president of ISTAM and also working as a chairman of admission committee of 5 years
integrated M.Sc. programme running at SVNIT, Surat. He organised around 2 international conferences, 4
STTP and Finishing school and delivered 10 invited talks at various national and international conference.
He has published around 40 research papers in various reputed journals at national and international level.
Manoj Mehta is Professor and Head of Department of Applied Mathematics &
Humanities, Sardar Vallabhbhai National Institute of Technology, Surat in INDIA. He
obtained Ph.D in 1977 from South Gujarat University in mathematics. His areas of
specialization are Fluid flow through porous media, Advanced Analytical & Numerical
Techniques. He has 39 years of teaching experience at undergraduate & post Graduate
level. He has supervised 20 Ph. D and one M. Phil student. He organised number of conferences, seminars
and workshops and delivered around 80 invited talks at various national and international conferences. He
has published over 125 research paper in well reputed journals of national and international level and
visited around 06 countries for academic purpose. He is member of many reputed societies like Indian
society of theoretical and applied mechanics (ISTAM), Indian Mathematical Society (IMS), Indian Society
for Technical Education (ISTE), Gujarat Ganit Mandal (GGM) and Indian Society of Bio Mechanics
(ISBM). He has written around 07 books at undergraduate and postgraduate level in mathematics.